geoms

The EasyFEA/geoms/ module in EasyFEA provides essential tools for creating and managing _Geom objects. These geometric objects are used to construct Mesh using the Mesher.

With this module, you can construct:

Point([x, y, z, isOpen, r])	Point class.
Points(points[, meshSize, isHollow, isOpen])	Points class.
Domain(pt1, pt2[, meshSize, isHollow])	Domain (2d or 3d domain) class.
Line(pt1, pt2[, meshSize, isOpen])	Line class.
Circle(center, diam[, meshSize, isHollow, ...])	Circle class.
CircleArc(pt1, pt2[, center, R, P, ...])	CircleArc class.
Contour(geoms[, isHollow, isOpen])	Contour class.

Once the geometric objects are created, you can manipulate them using copy(), Translate(), Rotate(), or Symmetry() (see the example for details).

Creating a Line

from EasyFEA.Geoms import Line

line = Line((0,0), (1,1))
line.Plot()

(Source code, png, hires.png, pdf)

Creating a Domain/Box

2D Domain

from EasyFEA.Geoms import Domain

domain = Domain((0, 0), (1, 1))
domain.Plot()

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3D Domain

from EasyFEA.Geoms import Domain

domain = Domain((0, 0, 0), (1, 1, 1))
domain.Plot()

(Source code, png, hires.png, pdf)

Creating a Circle

from EasyFEA.Geoms import Circle

circle = Circle((0, 0), 1.0)
circle.Plot()

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Using an axis normal to the circle

from EasyFEA.Geoms import Circle

circle = Circle((0, 0), 1.0, n=(0.5, 0.5, 0.5))
circle.Plot()

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Creating a CircleArc

From 2 Point and a Center

from EasyFEA.Geoms import CircleArc

circleArc = CircleArc((1, 0), (0, 1), center=(0, 0))
circleArc.Plot()

(Source code, png, hires.png, pdf)

From 2 Point and a Radius

from EasyFEA.Geoms import CircleArc

circleArc = CircleArc((1, 0), (0, 1), R=0.5)
circleArc.Plot()

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From 2 Point and a Point

from EasyFEA.Geoms import CircleArc

circleArc = CircleArc((1, 0), (0, 1), P=(0.8, 0.8))
circleArc.Plot()

(Source code, png, hires.png, pdf)

Creating a Contour from Points

from EasyFEA.Geoms import Points

contour = Points([(0, 0), (1,0), (1,1), (0,1)]).Get_Contour()
contour.Plot()

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Add a fillet

from EasyFEA.Geoms import Point, Points

contour = Points([Point(0, 0, r=0.5), (1,0), (1,1), (0,1)]).Get_Contour()
contour.Plot()

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from EasyFEA.Geoms import Point, Points

contour = Points([Point(0, 0, r=-0.5), (1,0), (1,1), (0,1)]).Get_Contour()
contour.Plot()

(Source code, png, hires.png, pdf)

Creating a Contour with Line, CircleArc and Points

from EasyFEA.Geoms import Line, CircleArc, Points, Contour

line = Line((0, 0), (1, 0))
points = Points([(1,0), (1.5, 0.5), (1, 1)])
circleArc = CircleArc((1,1), (0,0), center=(1,0))
contour = Contour([line, points, circleArc])
contour.Plot()

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Manipulate a _Geom object using the copy(), Translate(), Rotate(), and Symmetry() functions

from EasyFEA.Geoms import Points

contour1 = Points([(0,0), (1,0), (1,1), (0,1)]).Get_Contour()
contour2 = contour1.copy(); contour2.Translate(dx=4)
contour3 = contour2.copy(); contour3.Rotate(90, (0,0), (0,0,1))
contour4 = contour3.copy(); contour4.Symmetry((0,0), (0,1))

ax = contour1.Plot_Geoms([contour1, contour2, contour3, contour4], plotPoints=False)
ax.legend(["contour1", "contour2", "contour3", "contour4"])

(Source code, png, hires.png, pdf)

Detailed geoms API

Module containing the geometric classes used to build meshes.

Module containing the geometric functions used to build meshes.

EasyFEA.geoms.Angle_Between(a, b)

Computes the angle between vectors a and b (rad). https://math.stackexchange.com/questions/878785/how-to-find-an-angle-in-range0-360-between-2-vectors

Return type:

float

EasyFEA.geoms.AsCoords(value)

Returns value as a 3D vector

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

EasyFEA.geoms.AsPoint(coords)

Returns coords as a point.

Return type:

Point

class EasyFEA.geoms.Circle(center, diam, meshSize=0.0, isHollow=True, isOpen=False, n=(0, 0, 1))

Bases: _Geom

Circle class.

Get_Contour()

Creates the contour object associated with the circle

Get_coord_for_plot()

Returns lines and points coordinates for plotting.

Returns:

Lines and points coordinates as NumPy arrays.

Return type:

tuple of ndarray

__NCircle = 39

_abc_impl = <_abc._abc_data object>

property diam: float

circle’s diameter

property length: float

circle perimeter

property n: ndarray[tuple[Any, ...], dtype[Any]] | Collection[int | float]

axis normal to the circle

class EasyFEA.geoms.CircleArc(pt1, pt2, center=None, R=None, P=None, meshSize=0.0, n=(0, 0, 1), isOpen=False, coef=1)

Bases: _Geom

CircleArc class.

Get_coord_for_plot()

Returns lines and points coordinates for plotting.

Returns:

Lines and points coordinates as NumPy arrays.

Return type:

tuple of ndarray

__NCircleArc = 242

_abc_impl = <_abc._abc_data object>

property angle

circular arc angle [rad]

center

Point at the center of the arc.

property length: float

circular arc length

property n: ndarray[tuple[Any, ...], dtype[Any]] | Collection[int | float]

axis normal to the circle arc

pt1

Starting point of the arc.

pt2

Ending point of the arc.

pt3

Midpoint of the circular arc.

property r

circular arc radius

EasyFEA.geoms.Circle_Coords(coord, R, n)

Returns center from coordinates a radius and and a vector normal to the circle.

return center

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

EasyFEA.geoms.Circle_Triangle(p1, p2, p3)

Returns triangle’s center for the circumcicular arc formed by 3 points.

returns center

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

class EasyFEA.geoms.Contour(geoms, isHollow=True, isOpen=False)

Bases: _Geom

Contour class.

Get_coord_for_plot()

Returns lines and points coordinates for plotting.

Returns:

Lines and points coordinates as NumPy arrays.

Return type:

tuple of ndarray

__NContour = 111

_abc_impl = <_abc._abc_data object>

property length: float

class EasyFEA.geoms.Domain(pt1, pt2, meshSize=0.0, isHollow=True)

Bases: _Geom

Domain (2d or 3d domain) class.

Get_coord_for_plot()

Returns lines and points coordinates for plotting.

Returns:

Lines and points coordinates as NumPy arrays.

Return type:

tuple of ndarray

__NDomain = 266

_abc_impl = <_abc._abc_data object>

EasyFEA.geoms.Fillet(P0, P1, P2, r)

Computes fillet in a corner P0.

returns A, B, C

Parameters:

• P0 (_types.AnyArray) – coordinates of point with radius

• P1 (_types.AnyArray) – coordinates before P0 coordinates

• P2 (_types.AnyArray) – coordinates after P0 coordinates

• r (float) – radius (or fillet) at point P0

Returns:

coordinates calculated to construct the radius

Return type:

tuple[_types.FloatArray, _types.FloatArray, _types.FloatArray]

EasyFEA.geoms.Jacobian_Matrix(i, k)

Computes the Jacobian matrix to transform local coordinates (i,j,k) to global (x,y,z) coordinates.

p(x,y,z) = J • p(i,j,k) and p(i,j,k) = inv(J) • p(x,y,z)

ix jx kx

iy jy ky

iz jz kz

Parameters:

• i (_types.Coords) – i vector

• k (_types.Coords) – k vector

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

class EasyFEA.geoms.Line(pt1, pt2, meshSize=0.0, isOpen=False)

Bases: _Geom

Line class.

Get_coord_for_plot()

Returns lines and points coordinates for plotting.

Returns:

Lines and points coordinates as NumPy arrays.

Return type:

tuple of ndarray

__NLine = 845

_abc_impl = <_abc._abc_data object>

static distance(pt1, pt2)

Computes the distance between two points.

Return type:

float

static get_unitVector(pt1, pt2)

Creates the unit vector between two points.

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property length: float

distance between the two points of the line

property unitVector: ndarray[tuple[Any, ...], dtype[floating]]

The unit vector for the two points on the line (p2-p1)

EasyFEA.geoms.Normalize(array)

Must be a vector or matrix.

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

class EasyFEA.geoms.Point(x=0.0, y=0.0, z=0.0, isOpen=False, r=0.0)

Bases: object

Point class.

Check(coord)

Checks if coordinates are identical

Return type:

bool

PointALike

alias of Point | ndarray[tuple[Any, …], dtype[Any]] | Collection[int | float]

Rotate(theta, center=(0, 0, 0), direction=(0, 0, 1))

Rotates the point with around an axis.

Parameters:

• theta (float) – rotation angle [deg]

• center (_types.Coords, optional) – rotation center, by default (0,0,0)

• direction (_types.Coords, optional) – rotation direction, by default (0,0,1)

Return type:

None

Symmetry(point=(0, 0, 0), n=(1, 0, 0))

Symmetrizes the point coordinates with a plane.

Parameters:

• point (_types.Coords, optional) – a point belonging to the plane, by default (0,0,0)

• n (_types.Coords, optional) – normal to the plane, by default (1,0,0)

Return type:

None

Translate(dx=0.0, dy=0.0, dz=0.0)

Translates the point.

Return type:

None

property coord: ndarray[tuple[Any, ...], dtype[floating]]

[x,y,z] coordinates

copy()

isOpen: bool

point is open

r: float

radius used for fillet

property x: float

x coordinate

property y: float

y coordinate

property z: float

z coordinate

class EasyFEA.geoms.Points(points, meshSize=0.0, isHollow=True, isOpen=False)

Bases: _Geom

Points class.

Get_Contour()

Creates a contour from the points.

Creates a fillet if a point has a radius which is not 0.

Get_coord_for_plot()

Returns lines and points coordinates for plotting.

Returns:

Lines and points coordinates as NumPy arrays.

Return type:

tuple of ndarray

__NPoints = 59

_abc_impl = <_abc._abc_data object>

property length: float

pt1

First point

pt2

Last point

EasyFEA.geoms.Points_Intersect_Circles(circle1, circle2)

Computes the coordinates at the intersection of the two circles (i,3).

This only works if they’re on the same plane.

Parameters:

• circle1 (Circle) – circle 1

• circle2 (Circle) – circle 2

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

EasyFEA.geoms.Rotate(coord, theta, center=(0, 0, 0), direction=(0, 0, 1))

Rotates the coordinates arround a specified center and axis.

Parameters:

• coord (_types.AnyArray) – coordinates to rotate (n,3)

• theta (float) – rotation angle [deg]

• center (_types.Iterable, optional) – rotation center, by default (0,0,0)

• direction (_types.Iterable, optional) – rotation direction, by default (0,0,1)

Returns:

rotated coordinates

Return type:

_types.FloatArray

EasyFEA.geoms.Symmetry(coord, point=(0, 0, 0), n=(1, 0, 0))

Symmetrizes coordinates with a plane.

Parameters:

• coord (_types.AnyArray) – coordinates that we want to symmetrise

• point (tuple, optional) – a point belonging to the plane, by default (0,0,0)

• n (tuple, optional) – normal to the plane, by default (1,0,0)

Returns:

the new coordinates

Return type:

_types.FloatArray

EasyFEA.geoms.Translate(coord, dx=0.0, dy=0.0, dz=0.0)

Translates the coordinates.

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

class EasyFEA.geoms._Geom(points, meshSize, name, isHollow, isOpen)

Bases: ABC

Geometric class.

GeomCompatible

alias of _Geom | Domain | Circle | Points | Contour

abstractmethod Get_coord_for_plot()

Returns lines and points coordinates for plotting.

Returns:

Lines and points coordinates as NumPy arrays.

Return type:

tuple of ndarray

Mesh_2D(inclusions=[], elemType=ElemType.TRI3, cracks=[], refineGeoms=[], isOrganised=False, additionalSurfaces=[], additionalLines=[], additionalPoints=[], folder='')

Creates a 2D mesh from a contour and inclusions that must form a closed plane surface.

Parameters:

• inclusions (list[Domain, Circle, Points, Contour], optional) – list of hollow and filled geom objects inside the domain

• elemType (ElemType, optional) – element type, by default “TRI3” [“TRI3”, “TRI6”, “TRI10”, “TRI15”, “QUAD4”, “QUAD8”, “QUAD9”]

• cracks (list[Line | Points | Contour | CircleArc]) – list of geom object used to create open or closed cracks

• refineGeoms (list[Domain|Circle|str], optional) – list of geom object for mesh refinement, by default []

• isOrganised (bool, optional) – mesh is organized, by default False

• additionalSurfaces (list[Domain, Circle, Points, Contour]) – additional surfaces that will be added to or removed from the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). Tip: if the mesh is not well generated, you can also give the inclusions.

• additionalLines (list[Union[Line,CircleArc]]) – additional lines that will be added to the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). WARNING: lines must be within the domain.

• additionalPoints (list[Point]) – additional points that will be added to the surfaces created by the contour and the inclusions. WARNING: points must be within the domain.

• folder (str, optional) – default mesh.msh folder, by default “” does not save the mesh

Returns:

Created mesh

Return type:

Mesh

Mesh_Extrude(inclusions=[], extrude=(0, 0, 1), layers=[], elemType=ElemType.TETRA4, cracks=[], refineGeoms=[], isOrganised=False, additionalSurfaces=[], additionalLines=[], additionalPoints=[], folder='')

Creates a 3D mesh by extruding a surface constructed from a contour and inclusions.

Parameters:

• inclusions (list[Domain, Circle, Points, Contour], optional) – list of hollow and filled geom objects inside the domain

• extrude (Coords, optional) – extrusion vector, by default [0,0,1]

• layers (list[int], optional) – layers in the extrusion, by default []

• elemType (ElemType, optional) – element type, by default “TETRA4” [“TETRA4”, “TETRA10”, “HEXA8”, “HEXA20”, “HEXA27”, “PRISM6”, “PRISM15”, “PRISM18”]

• cracks (list[Line | Points | Contour | CircleArc]) – list of geom object used to create open or closed cracks

• refineGeoms (list[Domain|Circle|str], optional) – list of geom object for mesh refinement, by default []

• isOrganised (bool, optional) – mesh is organized, by default False

• additionalSurfaces (list[Domain, Circle, Points, Contour]) – additional surfaces that will be added to or removed from the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). Tip: if the mesh is not well generated, you can also give the inclusions.

• additionalLines (list[Union[Line,CircleArc]]) – additional lines that will be added to the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). WARNING: lines must be within the domain.

• additionalPoints (list[Point]) – additional points that will be added to the surfaces created by the contour and the inclusions. WARNING: points must be within the domain.

• folder (str, optional) – default mesh.msh folder, by default “” does not save the mesh

Returns:

Created mesh

Return type:

Mesh

Mesh_Revolve(inclusions=[], axis=None, angle=360, layers=[30], elemType=ElemType.TETRA4, cracks=[], refineGeoms=[], isOrganised=False, additionalSurfaces=[], additionalLines=[], additionalPoints=[], folder='')

Creates a 3D mesh by rotating a surface along an axis.

Parameters:

• inclusions (list[Domain, Circle, Points, Contour], optional) – list of hollow and filled geom objects inside the domain

• axis (Line, optional) – revolution axis, by default Line((0, 0), (0, 1))

• angle (float|int, optional) – revolution angle in [deg], by default 360

• layers (list[int], optional) – layers in extrusion, by default [30]

• elemType (ElemType, optional) – element type, by default “TETRA4” [“TETRA4”, “TETRA10”, “HEXA8”, “HEXA20”, “HEXA27”, “PRISM6”, “PRISM15”, “PRISM18”]

• cracks (list[Line | Points | Contour | CircleArc]) – list of geom object used to create open or closed cracks

• refineGeoms (list[Domain|Circle|str], optional) – list of geom object for mesh refinement, by default []

• isOrganised (bool, optional) – mesh is organized, by default False

• additionalSurfaces (list[Domain, Circle, Points, Contour]) – additional surfaces that will be added to or removed from the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). Tip: if the mesh is not well generated, you can also give the inclusions.

• additionalLines (list[Union[Line,CircleArc]]) – additional lines that will be added to the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). WARNING: lines must be within the domain.

• additionalPoints (list[Point]) – additional points that will be added to the surfaces created by the contour and the inclusions. WARNING: points must be within the domain.

• folder (str, optional) – default mesh.msh folder, by default “” does not save the mesh

Returns:

Created mesh

Return type:

Mesh

Plot(ax=None, color='', name='', lw=None, ls=None, plotPoints=True)

Plots the geometry using Matplotlib.

Parameters:

• ax (matplotlib axis, optional) – Axis to plot on. If None, a new one is created.

• color (str, optional) – Line color.

• name (str, optional) – Label for the object.

• lw (float, optional) – Line width.

• ls (str, optional) – Line style.

• plotPoints (bool, optional) – If True, display the object’s defining points.

Returns:

The axis with the plotted geometry.

Return type:

Axes

static Plot_Geoms(geoms, ax=None, color='', name='', plotPoints=True, plotLegend=True)

Plots a list of geometric objects on the same axis.

Parameters:

• geoms (list of _Geom) – Geometries to plot.

• ax (matplotlib axis, optional) – Axis to use. If None, a new one is created.

• color (str, optional) – Line color.

• name (str, optional) – Label for the geometries.

• plotPoints (bool, optional) – Whether to plot defining points.

• plotLegend (bool, optional) – Whether to display the legend.

Returns:

The axis with the plotted geometries.

Return type:

Axes

Rotate(theta, center=(0, 0, 0), direction=(0, 0, 1))

Rotates the object coordinates around an axis.

Parameters:

• theta (float) – rotation angle in [deg]

• center (tuple, optional) – rotation center, by default (0,0,0)

• direction (tuple, optional) – rotation direction, by default (0,0,1)

Return type:

None

Symmetry(point=(0, 0, 0), n=(1, 0, 0))

Reflects the geometry with respect to a plane.

Parameters:

• point (Coords, optional) – A point on the reflection plane, by default (0, 0, 0).

• n (Coords, optional) – Normal vector of the plane, by default (1, 0, 0).

Return type:

None

Translate(dx=0.0, dy=0.0, dz=0.0)

Translates the geometry in 3D space.

Parameters:

• dx (float, optional) – Translation along the x-axis, by default 0.0.

• dy (float, optional) – Translation along the y-axis, by default 0.0.

• dz (float, optional) – Translation along the z-axis, by default 0.0.

Return type:

None

_abc_impl = <_abc._abc_data object>

property coord: ndarray[tuple[Any, ...], dtype[floating]]

Returns the coordinates of all points as a NumPy array.

copy()

isHollow: bool

Indicates whether the formed geometry is hollow.

isOpen: bool

Indicates whether the geometry is open, typically to model cracks.

meshSize: float

Element size used for meshing.

name: str

Name of the geometric object.

property points: list[Point]

The list of points used to build the geometric object.